The Calculus of Voting

Yesterday Jeff and I did our patriotic duty and voted. It was strange because not only was it a Thursday, but this election was a general election for county offices and a primary for state and national offices (e.g. state senator, governor, and U.S. senator). We used these newfangled electronic voting machines. As a computer scientist, I have some reservations about electronic voting machines, particularly systems that do not have a paper backup. I don't think we have the notoriously bad machines here, but I still feel skeptical about their fairness.

The security issues of electronic voting machines could be a post of its own, but of course I know that my vast audience comes here for the math, not the computer science. So instead I am going to discuss another way in which elections may not be completely fair, even assuming the votes are counted perfectly.

Ideally, we should use a voting scheme that selects the most preferred candidate as winner. This would be the candidate who always win in a run-off between himself/herself and any other candidate. The most preferred candidate is also known as the Condorcet winner. In a vote between two opponents, selecting the most preferred candidate is simple: people vote for their favorite candidate, and whoever gets the most votes wins. But for a race with more than two opponents, selecting the Condorcet winner is complicated, and sometimes, there may not be a Condorcet winner.

With the voting system that we use, unless one candidate is clearly preferred over all the others, the Condorcet winner may not win. In fact, the Condorcet loser may end up being selected. A classic example of this is the presidential election of 1860. This was the year that Abraham Lincoln was elected president. That year, there were four candidates: Lincoln, Breckinridge, Bell, and Douglas. Lincoln carried most of the northern states, and the three other candidates split the remaining states. Let's pretend for the sake of argument that the winner of the popular vote would win the election. (In reality, Lincoln got 40% of the popular vote but 59% of the electoral college, so in some sense he won the majority of the vote.) Here are the election results, in percentages:
Lincoln: 40%
Douglas: 29%
Breckinridge: 18%
Bell: 13%

So Lincoln received the most votes, but he did not receive the majority of votes. Would he have won a runoff against every opposing candidate? Probably not. Chances are, if the people who voted for Breckinridge or Bell had to pick between Lincoln and Douglas, 99.9% of them would have selected Douglas, so Douglas would have won that race 60% to 40%. Similarly, Lincoln supporters would have probably preferred Douglas over Breckinridge or Bell. Douglas was almost certainly the Condorcet winner of that election.

In yesterday's primaries, there were a plethora of candidates for nearly every position. In the Democratic race for United States Senator, there were five candidates. As it turns out, one candidate, Harold Ford, Jr., got 79% of the votes, so there is no doubt that he was the Condorcet winner of that primary. But for the Republican race, there were four candidates, with the highest vote-count going to Bob Corker with only 48% of the vote. (Is it just me, or does that name make you laugh?) Anyhow, it is possible, although unlikely, that Bob Corker was actually the least preferred candidate, despite winning the election. Let's call his opponents X, Y, and Z. It could be that all supporters of X would choose Y over Bob and Z over Bob, and all supporters of Y would choose X or Z over Bob, and all supporters of Z would choose X or Y over Bob. I'm not trying to pick on poor Bob Corker, but I think you get the idea. The point is, Bob's supporters could be very overzealous about him and only him, but the supporters of X, Y, and Z could prefer anyone but Bob. This means that Bob was actually the least preferred candidate, despite the fact that he got the most votes out of anyone.

How can we be sure that the Condorcet winner is always elected? As I hinted above, we can't always be sure that there is a Condorcet winner. We could have a three-candidate cycle, for example. If we have three people (1, 2, and 3) voting in an election between three candidates (A, B, and C), the voters might have the following preferences:
1: A > B > C
2: B > C > A
3: C > A > B

So, if we ran A vs B, A would win 2-1; A vs C, C would win 2-1; and B vs C, B would win 2-1. So no candidate wins all the runoffs they are in; therefore there is no Condorcet winner.

But, assuming there is a Condorcet winner, there is a simple way to find it, and that is by having the voters rank the candidates in order of preference. From this data, a computer can compute the outcomes of all the runoffs, and determine the Condorcet winner. If the Condorcet winner does not exist, there must be some sort of tie-breaking method agreed upon ahead of time so that the election will have a winner.

Unfortunately, Condorcet voting methods are rarely used. Even in places like Europe where there are multiple political parties, the instant-runoff or Borda count methods are usually used, and neither are guaranteed to elect the Condorcet winner (assuming it exists).

Personally, I would like to see Condorcet voting methods implemented in the American election system. Now that our voting is becoming computerized, I don't see a problem with implementing these methods. It would take a computer very little time to find the Condorcet winner. And if one candidate garners over 50% of the vote, no computations would be needed!

If you are interested in learning more about Condorcet voting methods, may I recommend the Wikipedia article on the topic. Also, I got my information on the election of 1860 from the Wikipedia article on the United States Presidential Election of 1860.